Is there a word or phrase for this? When you take a word and extend the sound of a letter - usually a vowel - to intensify it. Sometimes you might say Helloooo and stretch out the ‘o’ sound. What is this called?
It's called word elongation or word lengthening - (reference: The Atlantic magazine)
Evvvvverywherrrre, from instant messages to texts to tweets and even e‑mails, I see examples of what language watchers call word lengthening. The habit began among teens and 20-somethings, but it is no longer limited to them. Adults are adding o’s to their no’s, s’s to their yes’es, and i’s to their hi’s, to say nothing of a glut of exclamation points. In response to some recent news, my 60-something mom wrote, “LOVE IT AND YOU TOO!!!!” What is going on?
For the past five years, Sali Tagliamonte, a linguist at the University of Toronto, has been gathering digital-communications data from students. In analyzing nearly 4 million words, she’s found some interesting patterns. “This reduplication of letters, it’s not all crazy,” she told me. Certain vowels—o, a, and e—are the most-frequent candidates for multiplication. Words are most frequently elongated by two or three letters at a time. Elongations are common in instant messaging and texting, but less frequent in e-mail. (reference: The Atlantic magazine)
Methodology: Prosodic indicators (such as high pitch, prolonged duration, intensity, vowel quality, etc.) have long been know (Bolinger, 1965) as ways for a speaker to emphasize or accent an important word. The ways in which they are used in speech are the subject of ongoing linguistic research (see, for example, Calhoun 2010). In written text, many of these indicators are lost. However, there exist some orthographic conventions which are used to mark or substitute for prosody, including punctuation and typographic styling (italic, bold, and underlined text). In purely text-based domains, such as Twitter, styling is not always available, and is replaced by capitalization or other conventions (e.g., enclosing the word in asterisks). Additionally, the informal nature of the domain leads to an orthographic style which is much closer to the spoken form than in other, more formal, domains. In this work, we hypothesize that the commonly observed phenomenon of lengthening words by repeating letters is a substitute for prosodic em- phasis (increased duration or change of pitch). As such, it can be used as an indicator of important words and, in particular, ones that bear strong indication of sentiment. (ACL web)
You can find full details on ACL website here: https://www.aclweb.org/anthology/D11-1052/
It can be referred to simply as "adding letters to words" or more conveniently as "word lengthening" (ref.).